Staff: Mentor

The equation y2 + xy + x2 - 2 = 0 is quadratic in y, so you can use the Quadratic Formula to solve for y. (You'll get two values with the ±, meaning there are two functions of x.)

Your problem is to find the minimum value of D(x) = √(x2 + y2). Here you will need to substitute for y from the previous work, making D really a function of x alone.

Alternatively, you could find the minimum value of D2(x) = x2 + y2, again making the substitutions for y. Since there are two functions, you'll need to do the work for both of them. I don't know if there is any symmetry you can exploit to save work. The graph is an ellipse, but one that has been rotated.

1. The problem statement, all variables and given/known data
Find the points [itex]x^2 + xy + y^2 = 2[/itex] that are closest to the origin.

2. Relevant equations
Distance Formula

3. The attempt at a solution
I have to first solve this without using Lagrange Multipliers.

This is essentially an ellipse. So I first completed the square:

[itex]3/4\,{x}^{2}+ \left( y+1/2\,x \right) ^{2}=2[/itex]

I was thinking first I should separate [itex]x[/itex] and [itex]y[/itex], and use that in the distance formula, but I can't seem to isolate [itex]y[/itex], is my approach wrong?

As an alternative method, you could solve the problem as a constrained optimization, using a Lagrange multiplier approach. You want to minimize f(x,y) = x^2 + y^2 (the square of the distance) subject to g(x,y) = 0, where g(x,y) = x^2 + y^2 + x*y-2.